The overall sink strength of an inhomogeneous lossy medium. Part II: variational estimates

Abstract

Trial fields generated by trial polarizations are substituted into the classical variational principles for the model system introduced in Part I. The resulting expressions are simplified by discarding certain terms which vanish quadratically at the actual solution, to yield a ‘Hashin-Shtrikman’ variational principle for the system under consideration. Substitution of simple configuration-dependent trial polarizations yields expressions that are identical with those obtained from the QCA in Part I. This demonstrates that the self-consistent QCA has an interpretation in terms of a variational principle and, furthermore, that the self-consistent prescription of Willis (1983, 1984) in fact produces a bound. The opposite bound can also be obtained by choosing the sink parameter of the comparison medium to be that of the inclusions. Results are presented for two cases, both assuming Percus-Yevick statistics. The bounds are usefully close when k 1 2 k 2 2 = ;10 ; the self-consistent QCA lies between them but the simple Brailsford-Bullough estimate does not. The case k 1 2 k 2 2 = ;10 6 is also discussed, as a model for voids in a lossy matrix.